Asymptotic Behavior of the Stock Price Distribution Density and Implied Volatility in Stochastic Volatility Models

TL;DR

This paper analyzes the long-term behavior of stock price and volatility densities in stochastic volatility models, providing explicit formulas and asymptotic estimates that improve understanding of implied volatility.

Contribution

It derives explicit asymptotic formulas for densities and implied volatility in Stein-Stein and Heston models, with error estimates, advancing quantitative understanding of these models.

Findings

Explicit asymptotic formulas for density of squared volatility and stock price.

Sharp asymptotic formulas for implied volatility in both models.

Error estimates for the asymptotic expansions.

Abstract

We study the asymptotic behavior of distribution densities arising in stock price models with stochastic volatility. The main objects of our interest in the present paper are the density of time averages of the squared volatility process and the density of the stock price process in the Stein-Stein and the Heston model. We find explicit formulas for leading terms in asymptotic expansions of these densities and give error estimates. As an application of our results, sharp asymptotic formulas for the implied volatility in the Stein-Stein and the Heston model are obtained.